# Why do higher frequencies tend to have a smaller amplitude than low frequencies?

Complex waves found in nature tend to have downward sloping frequency distribution. This isn't to say the fundamental always has the highest level but the general trend is that higher overtones tend to have lower levels.

The simple answer is that energy is proportional to the square of frequency and square of amplitude and so increasing frequency must mean amplitude decreases, however that's assuming the energy of each mode is the same.

Ok so what about the equipartition theorem? When a system has multiple degrees of freedom, over time the energy will become evenly distributed among the modes. But this assumes the modes are coupled in a significant way. If we're talking about a superposition of sines like a plucked guitar string there should be very little coupling (an idealized string should have 0 coupling since the harmonics are orthogonal).

However taken from here

Whether the modes are orthogonal or not really only covers the question of whether they interact via linear processes, e.g. superposition. Orthogonal modes can still interact with each other nonlinearly, which is typically the case when one observes harmonics (assuming they are real harmonics and not spontaneously-excited modes that happen to be at twice the frequency of another mode). This most commonly takes the form of quadratic phase coupling, i.e. there are quadratic terms in the dependent variable in the governing equations. That allows energy exchange between otherwise normal modes. While a guitar strings is likely pretty close to ideal for small amplitudes, it would be fairly unusual for a real-world problem with all the aforementioned complications to not include some degree of nonlinearity at higher amplitudes.

Then there's the purely mathematical answer: the total energy of the vibration has to be finite. If we have an infinite number of possible modes of vibration you need some distribution of the energy between few of them and you get less and less energy left for higher ones (a la convergence).

Now this isn't just for instruments, it also happens with electrical signals. I found this answer regarding electrical signals

the high frequency components in signals are by passed by the capacitive and inductive effects of the line. The value of these reactive components is such that they alter the higher frequencies more than the lower ones.

Note: I'm not saying the that the lowest frequency has to be the loudest followed by the second harmonic, etc. A trumpet is a clear counter example to this. Just that the general trend is that the lower frequencies in a sound tend to dominate and the higher harmonics tend to decrease in amplitude the higher up you go.

In my searching I stumbled upon this

A general answer is that most physical systems are low pass filters. They attenuate high frequencies more than low frequencies. There are exceptions, but most things are like that.

This makes sense for attenuation since attenuation is determined by the inverse of frequency but I'm talking about why higher frequencies aren't even excited to any significant degree in the first place

of course the frequency spectrum of a wave is related to the shape of the wave but I think this might play a part in why we tend to not observe waves where the high frequencies have a high amplitude.

• Why do you think "the total energy of the vibration has to be finite." is a "purely mathematical answer"? Do you think that any physical process can have infinite energy? Apr 20, 2021 at 10:27